Volume 60
, Issue 1

2024

Dear Reader, I am proud and excited to announce that it is the 60th anniversary of Parabola!

The isoperimetric problem asks for the least-perimeter way to enclose a given volume. We numerically solve this problem for double, triple and quadruple bubbles in the plane with density $r^p$ for various $p > 0$, using Brakke’s Evolver.

What is an amplifier and how does it work? To answer these intriguing questions, I constructed a model of an amplifier.

A ray of light emanates at some angle from a corner of a square region and follows a path determined by its reflections off the walls of the square. We determine when the ray’s path is finite, and we compute its length in this case.

How much do financial management fees cost investors? This article studies fees charged annually as a percentage of Assets Under Management (AUM).

We will attempt to multiply like a Babylonian student and will derive beautiful sexagesimal approximations.

Wacław Sierpiński proved that there exist infinitely many odd integers $k$ such that numbers of the form $k\cdot 2^n + 1$ are never prime for any integer $n$. The values of $k$ with this property are called Sierpiński numbers. The Sierpiński Problem is to find the smallest Sierpiński number.

The solution formula to the quadratic equation $ax^2+bx+c=0$ is usually derived in textbooks by completing the square. This is very unnatural and potentially confusing for students. A more appropriate approach is given here.

We describe Vieta Jumping, a technique that was used to solve the notorious 1988 International Mathematical Olympiad’s Problem 6. We provide explanations, examples and visual representations, as well as other problems that can be solved by this technique.

It is a well-known estimate that, for small values $x \geq 0$ much smaller than 1, the linear function $x$ approximates $\ln(1 + x)$. Alas, this easy approximation does not hold on all of the interval $[0,1]$. A far better almost-linear approximation is presented in this article.

I consider primeless and single-prime intervals of any given length, and show easy ways in which to construct them.

Q1732 Suppose that the numbers $a_1,a_2,\ldots,a_n$ are equal to $1,2,\ldots,n$, but not necessarily in that order. Find the maximum possible value of

$\displaystyle S = \sum_{k=1}^n (k-a_k)^2$